Decay of Fourier transforms and analytic continuation of power-constructible functions
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866911045602246656 |
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| author | Comte, Georges Miller, Dan J. Servi, Tamara |
| author_facet | Comte, Georges Miller, Dan J. Servi, Tamara |
| contents | For a subfield K of C, we denote by C^K the category of algebras of functions defined on the globally subanalytic sets that are generated by all K-powers and logarithms of positively-valued globally subanalytic functions. For any function f in C^\K(R), we study links between holomorphic extensions of f and the decay of its Fourier transform F[f] by using tameness properties of the globally subanalytic functions from which f is constructed. We first prove a number of theorems about analytic continuation of functions in C^K, including the fact that f in C^K(R) extends meromorphically to C if and only if f is rational. We then characterize the exponential rate of decay of F[f] by the maximal width of a horizontal strip in the plane about the real axis to which f extends holomorphically. Finally, we show that F[f] is integrable if f is integrable and continuous. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_06142 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Decay of Fourier transforms and analytic continuation of power-constructible functions Comte, Georges Miller, Dan J. Servi, Tamara Algebraic Geometry 26B15, 14P15, 32B20, 42B20, 42A38 (Primary) For a subfield K of C, we denote by C^K the category of algebras of functions defined on the globally subanalytic sets that are generated by all K-powers and logarithms of positively-valued globally subanalytic functions. For any function f in C^\K(R), we study links between holomorphic extensions of f and the decay of its Fourier transform F[f] by using tameness properties of the globally subanalytic functions from which f is constructed. We first prove a number of theorems about analytic continuation of functions in C^K, including the fact that f in C^K(R) extends meromorphically to C if and only if f is rational. We then characterize the exponential rate of decay of F[f] by the maximal width of a horizontal strip in the plane about the real axis to which f extends holomorphically. Finally, we show that F[f] is integrable if f is integrable and continuous. |
| title | Decay of Fourier transforms and analytic continuation of power-constructible functions |
| topic | Algebraic Geometry 26B15, 14P15, 32B20, 42B20, 42A38 (Primary) |
| url | https://arxiv.org/abs/2507.06142 |